Abstract: We determine a class of rearrangements that admit a supporting tree. Thiscondition implies that the associated rearrangement operator has a boundedvector valued extension. We show that there exists a large subspace of $L^p$ onwhich a bounded rearrangement operator acts as an isomorphism. Thecombinatorial issues of these problems give rise to a two-person game, to beplayed with colored dyadic intervals. We determine winning strategies for eachof the players.